Two-torsion subgroups of some modular Jacobians

Abstract

We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus [Formula: see text], [Formula: see text] or [Formula: see text]. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the [Formula: see text]-torsion subgroup. Using [Formula: see text]-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton–Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the [Formula: see text]-torsion points. We demonstrate the practicality of our method by computing the [Formula: see text]-torsion of the modular Jacobians [Formula: see text] for [Formula: see text]. As a result of this we are able to verify the generalized Ogg conjecture for these values.